Science, Mathematics, & Computing

Permanent URI for this collection


Recent Submissions

Now showing 1 - 5 of 20
  • Item
    Analytic torsion and Faddeev-Popov ghosts
    (2002) McIntyre, Andrew
    The regularized determinant of the Laplacian on n-differentials on a hyperbolic Riemann surface is studied. The main result is an intrinsic characterization of the connection form for the determinant line bundle, endowed with the Quillen metric, over the Teichmüller space, in terms of the Green’s function of the Cauchy-Riemann operator. Further, an explicit series representation of that Green’s function, on a Schottky uniformization of the surface, is established. This is a rigorous version of physical heuristics due to Martinec and Verlinde & Verlinde, relating the determinant to the stress-energy tensor of Faddeev-Popov ghost fields on the Riemann surface. One corollary is a simpler proof of the rigorous hyperbolic Belavin-Knizhnik formula, due to Zograf and Takhtajan, which is an intrinsic characterization of the curvature form of the determinant line bundle with Quillen metric. Another corollary is a proof of an explicit holomorphic factorization formula for n = 1 and genus greater than 1, due to Zograf, which generalizes the well known formula for n = 1 and genus 1 relating the determinant of the Laplacian to the Dedekind eta function.
  • Item
    Holomorphic factorization of determinants of Laplacians on Riemann surfaces and a higher genus generalization of Kronecker’s first limit formula
    (Springer Verlag, 2006-09) McIntyre, Andrew; Takhtajan, Leon A.
    For a family of compact Riemann surfaces X_t of genus g>1 parametrized by the Schottky space S_g, we define a natural basis for the holomorphic n-differentials on X_t which varies holomorphically with t and generalizes the basis of normalized abelian differentials of the first kind for n=1. We introduce a holomorphic function F(n) on S_g which generalizes the classical product \prod(1-q^m)^2 appearing in the Dedekind eta function for n=1 and g=1. We prove a holomorphic factorization formula expressing the regularized determinant of the Laplacian as a product of |F(n)|^2, a holomorphic anomaly depending on the classical Liouville action (a Kahler potential of S_g), and the determinant of the Gram matrix of the natural basis. The factorization formula reduces to Kronecker's first limit formula when n=1 and g=1, and to Zograf's factorization formula for n=1 and g>1.
  • Item
    Holomorphic factorization of determinants of Laplacians using quasi-Fuchsian uniformization
    (Springer Verlag, 2006) McIntyre, Andrew; Teo, Lee-Peng
    For a quasi-Fuchsian group Γ with ordinary set Ω, and Δ_n the Laplacian on n-differentials on Γ\Ω, we define a notion of a Bers dual basis ϕ_1,…,ϕ_2_d for ker Δ_n. We prove that det Δ_n/det⟨ϕ_j,ϕ_k⟩, is, up to an anomaly computed by Takhtajan and the second author in (Commun. Math Phys 239(1-2):183–240, 2003), the modulus squared of a holomorphic function F(n), where F(n) is a quasi-Fuchsian analogue of the Selberg zeta function Z(n). This generalizes the D’Hoker–Phong formula det Δ_n=c_g,_nZ(n), and is a quasi-Fuchsian counterpart of the result for Schottky groups proved by Takhtajan and the first author in Analysis 16, 1291–1323, 2006.
  • Item
    Tau function and Chern–Simons invariant
    (Elsevier, 2014) McIntyre, Andrew; Park, Jinsung
    We define a Chern-Simons invariant for a certain class of infinite volume hyperbolic 3-manifolds. We then prove an expression relating the Bergman tau function on a cover of the Hurwitz space, to the lifting of the function F defined by Zograf on Teichmüller space, and another holomorphic function on the cover of the Hurwitz space which we introduce. If the point in cover of the Hurwitz space corresponds to a Riemann surface X, then this function is constructed from the renormalized volume and our Chern-Simons invariant for the bounding 3-manifold of X given by Schottky uniformization, together with a regularized Polyakov integral relating determinants of Laplacians on X in the hyperbolic and singular flat metrics. Combining this with a result of Kokotov and Korotkin, we obtain a similar expression for the isomonodromic tau function of Dubrovin. We also obtain a relation between the Chern-Simons invariant and the eta invariant of the bounding 3-manifold, with defect given by the phase of the Bergman tau function of X.
  • Item
    Strain localization on an oceanic detachment fault system, Atlantis Massif, 30 °N, Mid-Atlantic Ridge
    (Wiley American Geophysical Union, 2004-11) Schroeder, Timothy; John, Barbara E.
    Abstract: Microstructural observations, mineral chemistry, and the spatial distribution of deformation fabrics recorded in outcrop samples collected from Atlantis Massif, the active inside corner high at 30 °N, Mid- Atlantic Ridge, suggest that strain is localized near the subhorizontal domal surface hypothesized to be an exposed detachment fault. Deformation textures in peridotite and gabbro indicate that high-temperature (>500 °C) strain occurred via crystal-plastic flow and diffusive mass transfer. Low-temperature (<400 °C) shear zones containing brittle and semibrittle microboudinage textures in which tremolite, chlorite, and/or talc replace fractured serpentine or hornblende cut earlier formed high-temperature deformation fabrics in peridotite. Textures indicate strain was localized by cataclasis and reaction softening into zones of intense greenschist and subgreenschist grade metamorphism. Gabbro is only weakly deformed below amphibolite facies (<500° C), indicating that strain was partitioned into altered peridotite at low temperature. There is a clear relationship between deformation intensity and structural depth beneath the subhorizontal surface of the Massif. Discontinuous high-intensity crystal-plastic deformation fabrics are found at all structural depths (0–520 m) beneath the surface, indicating that high-temperature, granulite- and amphibolite-grade deformation was not localized in a single shear zone. In contrast, semibrittle and brittle low-temperature shear zones are concentrated less than 90 m structurally beneath the surface, and the most intensely brittlely deformed samples concentrated in the upper 10 m. Localization of brittle deformation fabrics near the upper surface of the massif supports the hypothesis that it is the exposed footwall of a detachment fault. The structural evolution of Atlantis Massif is therefore analogous to a continental metamorphic core complex. Strain was localized onto the fault by reaction-softening and fluid-assisted fracturing during greenschist- and subgreenschist-grade hydrothermal alteration of olivine, clinopyroxene, serpentine, and hornblende to tremolite, chlorite, and/or talc. Keywords: detachment fault; low-angle normal fault; megamullion; metamorphic core complex; oceanic core complex; slow-spreading ridges.