Browsing by Author "McIntyre, Andrew"
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ItemAnalytic torsion and Faddeev-Popov ghosts(2002) McIntyre, AndrewThe 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. ItemHolomorphic 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. ItemHolomorphic factorization of determinants of Laplacians using quasi-Fuchsian uniformization(Springer Verlag, 2006) McIntyre, Andrew; Teo, Lee-PengFor 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. ItemTau function and Chern–Simons invariant(Elsevier, 2014) McIntyre, Andrew; Park, JinsungWe 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.