The Helton-Howe Trace Formula for the Fock Space

The famous Helton-Howe trace formula was originally established for antisymmetric sums of Toeplitz operators on the Bergman space of the unit ball. This formula has been extended to weighted Bergman spaces and to the Hardy space in [20], and more recently it even has been extended to the Drury-Arveson space [24]. In this paper we show that an analogue of the Helton-Howe trace formula also holds in an unlikely place, the Fock space! We further show that our techniques not only work for the ordinary trace tr, as in the case of the Helton-Howe trace formula, but also for the Dixmier trace.