Hyperbolic three-string vertex

We begin developing tools to compute off-shell string amplitudes with the recently proposed hyperbolic string vertices of Costello and Zwiebach. Exploiting the relation between a boundary value problem for Liouville’s equation and a monodromy problem for a Fuchsian equation, we construct the local coordinates around the punctures for the generalized hyperbolic three-string vertex and investigate their various limits. This vertex corresponds to the general pants diagram with three boundary geodesics of unequal lengths. We derive the conservation laws associated with such vertex and perform sample computations. We note the relevance of our construction to the calculations of the higher-order string vertices using the pants decomposition of hyperbolic Riemann surfaces.

Uniformization of p-adic curves via Higgs–de Rham flows

Let k be an algebraic closure of a finite field of odd characteristic. We prove that for any rank two graded Higgs bundle with maximal Higgs field over a generic hyperbolic curve {X_{1}} defined over k, there exists a lifting X of the curve to the ring {W(k)} of Witt vectors as well as a lifting of the Higgs bundle to a periodic Higgs bundle over {X/W(k)} . These liftings give rise to a two-dimensional absolutely irreducible representation of the arithmetic fundamental group {\pi_{1}(X_{K})} of the generic fiber of X. This curve X and its associated representation is in close relation to the canonical curve and its associated canonical crystalline representation in the p-adic Teichmüller theory for curves due to S. Mochizuki. Our result may be viewed as an analogue of the Hitchin–Simpson’s uniformization theory of hyperbolic Riemann surfaces via Higgs bundles.

Finiteness theorems for holomorphic mapping from products of hyperbolic Riemann surfaces

We prove that the space of dominant/non-constant holomorphic mappings from a product of hyperbolic Riemann surfaces of finite type into certain hyperbolic manifolds that can be covered by a bounded domain is a finite set.