  If V is a bundle of Tate vector spaces over a base B, its determinantal gerbe
has a class C_1(V) in the second cohomology group of the sheaf of invertible
functions which can be seen as the Deligne cohomology H^3(B, Z(2)).
  An example of such a "Tate bundle" can be obtained from a finite rank vector
bundle E on the product of B and a punctured formal disk. Our main result
identifies the corresponding C_1(V) with the cohomological direct image of
ch_2(E), the second Chern character of E. It can be seen as a "local" version
of the Riemann-Roch-Grothendieck theorem for a family of curves.
  This theorem explains the results of Gorbounov, Malikov and Schechtman
relating ch_2 of the tangent bundle of an algebraic variety X to the existence
of a sheaf of chiral differential operators. To be precise, it implies that the
determinantal anomaly of the formal loop space of X is the transgression of
ch_2(TX).
