Relative entropy for compact Riemann surfaces - art. no. 084001

The relative entropy of the massive free bosonic field theory is studied on various compact Riemann surfaces as a universal quantity with physical sig nificance, in particular, Fur gravitational phenomena. The exact expression for the sphere is obtained, as well as its asymptotic series for large mas s and its Taylor series for small mass. One can also derive exact expressio ns for the torus but not for higher genus. However. the asymptotic behavior for large mass can always be established-up to a constant-with heat-kernel methods. It consists of an asymptotic series determined only by the curvat ure-and, hence, is common for homogeneous surfaces of genus higher than one -and exponentially vanishing corrections whose form is determined by the co ncrete topology. The coefficient of the logarithmic term in this series giv es the conformal anomaly.