Genus zero Gopakumar-Vafa type invariants for Calabi-Yau 4-folds

In analogy with the Gopakumar-Vafa conjecture on CY 3-folds, Klemm and Pandharipande defined GV type invariants on Calabi-Yau 4-folds using Gromov-Witten theory and conjectured their integrality. In this paper, we propose a sheaf-theoretic interpretation of their genus zero invariants using Donaldson-Thomas theory on CY 4-folds. More specifically, we conjecture genus zero GV type invariants are $\mathrm{DT_{4}}$ invariants for one-dimensional stable sheaves on CY 4-folds. Some examples are computed for both compact and non-compact CY 4-folds to support our conjectures. We also propose an equivariant version of the conjectures for local curves and verify them in certain cases.Comment: 30 page

Toric Representation and Positive Cone of Picard Group and Deformation Space in Mirror Symmetry of Calabi-Yau Hypersurfaces in Toric Varieties

We derive the combinatorial representations of Picard group and deformation space of anti-canonical hypersurfaces of a toric variety using techniques in toric geometry. The mirror cohomology correspondence in the context of mirror symmetry is established for a pair of Calabi-Yau (CY) ${\sf n}$-spaces in toric varieties defined by reflexive polytopes for an arbitrary dimension ${\sf n}$. We further identify the Kahler cone of the toric variety and degeneration cone of CY hypersurfaces, by which the Kahler cone and degeneration cone for a mirror CY pair are interchangeable under mirror symmetry. In particular, different degeneration cones of a CY 3-fold are corresponding to flops of its mirror 3-fold.Comment: Latex 29 pag

Changes in back fat thickness during late gestation predict colostrum yield in sows

Directing protein and energy sources towards lactation is crucial to optimise milk production in sows but how this influences colostrum yield (CY) remains unknown. The aim of this study was to identify associations between CY and the sow’s use of nutrient resources. We included 37 sows in the study that were all housed, fed and managed similarly. Parity, back fat change (ΔBF), CY and performance parameters were measured. We obtained sow serum samples 3 to 4 days before farrowing and at D1 of lactation following overnight fasting. These were analysed for non-esterified fatty acids (NEFA), urea, creatinine, (iso) butyrylcarnitine (C4) and immunoglobulins G (IgG) and A (IgA). The colostrum samples collected 3, 6 and 24 h after the birth of the first piglet were analysed for their nutrient and immunoglobulins content. The technical parameters associated with CY were parity group (a; parities 1 to 3 =value 0 v. parities 4 to 7 =value 1) and ΔBF D85-D109 of gestation (mm) (b): CY (g) =4290–842a–113b. ( R2=0.41, P<0.001). The gestation length ( P<0.001) and the ΔBF between D109 and D1 of lactation (P=0.050) were identified as possible underlying factors of the parity group. The metabolic parameters associated with CY were C4 at 3 to 4 days before farrowing (a), and 10logC4 (b) and 10logNEFA (c) at D1 of lactation: CY ( g) =3582–1604a+1007b− 922c ( R2=0.39, P=0.001). The colostrum composition was independent of CY. The negative association between CY and ΔBF D85-D109 of gestation could not be further explained based on our data. Sows that were catabolic 1 week prior to farrowing seemed unable to produce colostrum to their full potential. This was especially the case for sows with parities 4 to 7, although they had a similar feed intake, litter birth weight and colostrum composition compared with parities 1 to 3 sows. In conclusion, this study showed that parity and the use of body fat and protein reserves during late gestation were associated with CY, indicating that proper management of the sow’s body condition during late gestation could optimise the intrinsic capacity of the sow’s CY

Period Integrals of CY and General Type Complete Intersections

We develop a global Poincar\'e residue formula to study period integrals of families of complex manifolds. For any compact complex manifold $X$ equipped with a linear system $V^*$ of generically smooth CY hypersurfaces, the formula expresses period integrals in terms of a canonical global meromorphic top form on $X$. Two important ingredients of our construction are the notion of a CY principal bundle, and a classification of such rank one bundles. We also generalize our construction to CY and general type complete intersections. When $X$ is an algebraic manifold having a sufficiently large automorphism group $G$ and $V^*$ is a linear representation of $G$, we construct a holonomic D-module that governs the period integrals. The construction is based in part on the theory of tautological systems we have developed in the paper \cite{LSY1}, joint with R. Song. The approach allows us to explicitly describe a Picard-Fuchs type system for complete intersection varieties of general types, as well as CY, in any Fano variety, and in a homogeneous space in particular. In addition, the approach provides a new perspective of old examples such as CY complete intersections in a toric variety or partial flag variety.Comment: An erratum is included to correct Theorem 3.12 (Uniqueness of CY structure

Effect of ovariectomy on the progression of chronic kidney disease-mineral bone disorder (CKD-MBD) in female Cy/+ rats

Male Cy/+ rats have shown a relatively consistent pattern of progressive kidney disease development that displays multiple key features of late stage chronic kidney disease-mineral bone disorder (CKD-MBD), specifically the development of cortical bone porosity. However, progression of disease in female Cy/+ rats, assessed in limited studies, is more heterogeneous and to date has failed to show development of the CKD-MBD phenotype, thus limiting their use as a practical model of progressive CKD-MBD. Animal and human studies suggest that estrogen may be protective against kidney disease in addition to its established protective effect on bone. Therefore, in this study, we aimed to determine the effect of ovariectomy (OVX) on the biochemical and skeletal manifestations of CKD-MBD in Cy/+ female rats. We hypothesized that OVX would accelerate development of the biochemical and skeletal features of CKD-MBD in female Cy/+ rats, similar to those seen in male Cy/+ rats. Female Cy/+ rats underwent OVX (n = 8) or Sham (n = 8) surgery at 15 weeks of age. Blood was collected every 5 weeks post-surgery until 35 weeks of age, when the rats underwent a 4-day metabolic balance, and the tibia and final blood were collected at the time of sacrifice. OVX produced the expected changes in trabecular and cortical parameters consistent with post-menopausal disease, and negative phosphorus balance compared with Sham. However, indicators of CKD-MBD were similar between OVX and Sham (similar kidney weight, plasma blood urea nitrogen, creatinine, creatinine clearance, phosphorus, calcium, parathyroid hormone, and no cortical porosity). Contrary to our hypothesis, OVX did not produce evidence of development of the CKD-MBD phenotype in female Cy/+ rats

Graded Quivers, Generalized Dimer Models and Toric Geometry

The open string sector of the topological B-model model on CY $(m+2)$-folds is described by $m$-graded quivers with superpotentials. This correspondence extends to general $m$ the well known connection between CY $(m+2)$-folds and gauge theories on the worldvolume of D$(5-2m)$-branes for $m=0,\ldots, 3$. We introduce $m$-dimers, which fully encode the $m$-graded quivers and their superpotentials, in the case in which the CY $(m+2)$-folds are toric. Generalizing the well known $m=1,2$ cases, $m$-dimers significantly simplify the connection between geometry and $m$-graded quivers. A key result of this paper is the generalization of the concept of perfect matching, which plays a central role in this map, to arbitrary $m$. We also introduce a simplified algorithm for the computation of perfect matchings, which generalizes the Kasteleyn matrix approach to any $m$. We illustrate these new tools with a few infinite families of CY singularities.Comment: 54 pages, 6 figure