Some ball quotients with a Calabi--Yau model

Recently we determined explicitly a Picard modular variety of general type. On the regular locus of this variety there are holomorphic three forms which have been constructed as Borcherds products. Resolutions of quotients of this variety, such that the zero divisors are in the branch locus, are candidates for Calabi-Yau manifolds. Here we treat one distinguished example for this. In fact we shall recover a known variety given by the equations $X_0X_1X_2=X_3X_4X_5, \,\, X_0^3+X_1^3+X_2^3=X_3^3+X_4^3+X_5^3.$ as a Picard modular variety. This variety has a projective small resolution which is a rigid Calabi-Yau manifold ($h^{12}=0$) with Euler number $72$

The modular variety of hyperelliptic curves of genus three

The modular variety of non singular and complete hyperelliptic curves with level-two structure of genus 3 is a 5-dimensional quasi projective variety which admits several standard compactifications. The first one, X, comes from the realization of this variety as a sub-variety of the Siegel modular variety of level two and genus three .We will be to describe the equations of X in a suitable projective embedding and its Hilbert function. It will turn out that X is normal. A further model comes from geometric invariant theory using so-called semistable degenerated point configurations in (P^1)^8 . We denote this GIT-compactification by Y. The equations of this variety in a suitable projective embedding are known. This variety also can by identified with a Baily-Borel compactified ball-quotient. We will describe these results in some detail and obtain new proofs including some finer results for them. We have a birational map between Y and X . In this paper we use the fact that there are graded algebras (closely related to algebras of modular forms) A,B such that X=proj(A) and Y=proj(B). This homomorphism rests on the theory of Thomae (19th century), in which the thetanullwerte of hyperelliptic curves have been computed. Using the explicit equations for $A,B$ we can compute the base locus of the map from Y to X. Blowing up the base locus and the singularity of Y, we get a dominant, smooth model {\tilde Y}. We will see that {\tilde Y} is isomorphic to the compactification of families of marked projective lines (P^1,x_1,...,x_8), usually denoted by {\bar M_{0,8}}. There are several combinatorial similarities between the models X and Y. These similarities can be described best, if one uses the ball-model to describe Y.Comment: 39 page

Some Siegel threefolds with a Calabi-Yau model II

In the paper [FSM] we described some Siegel modular threefolds which admit a Calabi-Yau model. Using a different method we give in this paper an enlarged list of such varieties that admits a Calabi-Yau model in the following weak sense: there exists a desingularization in the category of complex spaces of the Satake compactification which admits a holomorphic three-form without zeros and whose first Betti number vanishes Basic for our method is the paper [GN] of van Geemen and Nygaard.Comment: 23 pages, no figure

Classical theta constants vs. lattice theta series, and super string partition functions

Recently, various possible expressions for the vacuum-to-vacuum superstring amplitudes has been proposed at genus $g=3,4,5$. To compare the different proposals, here we will present a careful analysis of the comparison between the two main technical tools adopted to realize the proposals: the classical theta constants and the lattice theta series. We compute the relevant Fourier coefficients in order to relate the two spaces. We will prove the equivalence up to genus 4. In genus five we will show that the solutions are equivalent modulo the Schottky form and coincide if we impose the vanishing of the cosmological constant.Comment: 21 page

The vanishing of two-point functions for three-loop superstring scattering amplitudes

In this paper we show that the two-point function for the three-loop chiral superstring measure ansatz proposed by Cacciatori, Dalla Piazza, and van Geemen vanishes. Our proof uses the reformulation of ansatz in terms of even cosets, theta functions, and specifically the theory of the $\Gamma_{00}$ linear system on Jacobians introduced by van Geemen and van der Geer. At the two-loop level, where the amplitudes were computed by D'Hoker and Phong, we give a new proof of the vanishing of the two-point function (which was proven by them). We also discuss the possible approaches to proving the vanishing of the two-point function for the proposed ansatz in higher genera

Exploring the Heidelberg Retinal Tomograph 3 diagnostic accuracy across disc sizes and glaucoma stages: a multicenter study

To investigate and compare the diagnostic accuracy of the Heidelberg Retinal Tomograph 3 (HRT3) diagnostic algorithms and establish whether they are affected by optic disc size and glaucoma severity

Extending the Belavin-Knizhnik "wonderful formula" by the characterization of the Jacobian

A long-standing question in string theory is to find the explicit expression of the bosonic measure, a crucial issue also in determining the superstring measure. Such a measure was known up to genus three. Belavin and Knizhnik conjectured an expression for genus four which has been proved in the framework of the recently introduced vector-valued Teichmueller modular forms. It turns out that for g>3 the bosonic measure is expressed in terms of such forms. In particular, the genus four Belavin-Knizhnik "wonderful formula" has a remarkable extension to arbitrary genus whose structure is deeply related to the characterization of the Jacobian locus. Furthermore, it turns out that the bosonic string measure has an elegant geometrical interpretation as generating the quadrics in P^{g-1} characterizing the Riemann surface. All this leads to identify forms on the Siegel upper half-space that, if certain conditions related to the characterization of the Jacobian are satisfied, express the bosonic measure as a multiresidue in the Siegel upper half-space. We also suggest that it may exist a super analog on the super Siegel half-space.Comment: 15 pages. Typos corrected, refs. and comments adde

Protease-activated receptor 2 sensitizes the capsaicin receptor transient receptor potential vanilloid receptor 1 to induce hyperalgesia

Inflammatory proteases (mast cell tryptase and trypsins) cleave protease-activated receptor 2 (PAR2) on spinal afferent neurons and cause persistent inflammation and hyperalgesia by unknown mechanisms. We determined whether transient receptor potential vanilloid receptor 1 (TRPV1), a cation channel activated by capsaicin, protons, and noxious heat, mediates PAR2-induced hyperalgesia. PAR2 was coexpressed with TRPV1 in small- to medium-diameter neurons of the dorsal root ganglia (DRG), as determined by immunofluorescence. PAR2 agonists increased intracellular [Ca2+] ([Ca2+]i) in these neurons in culture, and PAR2-responsive neurons also responded to the TRPV1 agonist capsaicin, confirming coexpression of PAR2 and TRPV1. PAR2 agonists potentiated capsaicin-induced increases in [Ca2+]i in TRPV1-transfected human embryonic kidney (HEK) cells and DRG neurons and potentiated capsaicin-induced currents in DRG neurons. Inhibitors of phospholipase C and protein kinase C (PKC) suppressed PAR2-induced sensitization of TRPV1-mediated changes in [Ca2+]i and TRPV1 currents. Activation of PAR2 or PKC induced phosphorylation of TRPV1 in HEK cells, suggesting a direct regulation of the channel. Intraplantar injection of a PAR2 agonist caused persistent thermal hyperalgesia that was prevented by antagonism or deletion of TRPV1. Coinjection of nonhyperalgesic doses of PAR2 agonist and capsaicin induced hyperalgesia that was inhibited by deletion of TRPV1 or antagonism of PKC. PAR2 activation also potentiated capsaicin-induced release of substance P and calcitonin gene-related peptide from superfused segments of the dorsal horn of the spinal cord, where they mediate hyperalgesia. We have identified a novel mechanism by which proteases that activate PAR2 sensitize TRPV1 through PKC. Antagonism of PAR2, TRPV1, or PKC may abrogate protease-induced thermal hyperalgesia

Comments on Anomaly Cancellations by Pole Subtractions and Ghost Instabilities with Gravity

We investigate some aspects of anomaly cancellation realized by the subtraction of an anomaly pole, stressing on some of its properties in superspace. In a local formulation these subtractions can be described in terms of a physical scalar, an axion and related ghosts. They appear to be necessary for the unitarization of the theory in the ultraviolet, but they may generate an infrared instability of the corresponding effective action, signalled by ghost condensation. In particular the subtraction of the superanomaly multiplet by a pole in superspace is of dubious significance, due to the different nature of the chiral and conformal anomalies. In turn, this may set more stringent constraints on the coupling of supersymmetric theories to gravity.Comment: 18 pages. Revised version. To appear in "Classical and Quantum Gravity