Tau-functions on Hurwitz Spaces

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Solution of matrix Riemann-Hilbert problems with quasi-permutation monodromy matrices (2003, unknown): “The proofs of both theorems are contained in [ 16 ]; they are based on the properties of Dirichlet action corresponding to the metric dλdλ in genus zero, and the metric ww in genus 1.”
Tau-functions on spaces of Abelian differentials and higher genus generalizations of Ray-Singer formula (2004, methodology): “FA α )z has no jumps on the cuts a βand b , β = 1,...,g.) By means of asymptotical expansions of the integrands at the zeroes Pm of w (Lemma 1 plays here a central role; cf. the proof of Theorem 4 in [23]) one gets the relation 1 2i ( XM m=1 I Pm ) {FA α |φ z| 2d¯z−(FA α ) φ dz +(FαA) φ¯d¯z} = (3.28”
Invariants of algebraic curves and topological expansion (2007, background): “auchy residue formula: for any meromorphic function f(p) we have: f(p) = − Res q1→p dSq1,q2 (p)f(q 1). (3-38) 16 3.1.11 Bergmann tau function The Bergmann τ-function τBx was introduced and studied in [61, 62, 37], it is such that ∂ln(τBx) ∂x(ai) = Res p→a i B(p,p) dx(p) (3-39) It is well deﬁned beca”