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Conformal field theory and the geometry of second quantization.

We examine the holomorphic representation of the quantum mechanics of two-dimensional Weyl-Majorana fermions coupled to a static background geometry. We give an explicit construction of the states and consider their parallel transport on the space of background geometries. We use this parallel transport to exhibit the anomalous transformation properties of quantum states under diffeomorphisms of the background metric and show that the central extension of the diffeomorphism algebra is given by the adiabatic curvature of the vacuum state. We show that the adiabatic effective action exhibits a gravitational anomaly which is also directly related to the adiabatic curvature. We present explicit formulas for Berry's connection and curvature for the quantum states of the theory.