Analytic Torsion for Surfaces with Cusps I: Compact Perturbation Theorem and Anomaly Formula

We define the analytic torsion associated with a Riemann surface endowed with a metric having Poincaré-type singularities in the neighborhood of a finite number of points and a Hermitian vector bundle with at most logarithmic singularities at those points, coming from the metric on the negative power of the canonical line bundle twisted by the divisor of the points. Then we provide a relation between this analytic torsion and the Ray–Singer analytic torsion of the compactified surface. From this relation we then establish the anomaly formula, which describes how the analytic torsion changes under the change of the metric on the surface and on the vector bundle.