  The half filled Landau level is expected to be approximately particle-hole
symmetric, which requires an extension of the Halperin-Lee-Read (HLR) theory of
the compressible state observed at this filling. Recent work indicates that,
when particle-hole symmetry is preserved, the composite Fermions experience a
quantized $\pi$-Berry phase upon winding around the composite Fermi-surface,
analogous to Dirac fermions at the surface of a 3D topological insulator. In
contrast, the effective low energy theory of the composite fermion liquid
originally proposed by HLR lacks particle-hole symmetry and has vanishing Berry
phase. In this paper, we explain how thermoelectric transport measurements can
be used to test the Dirac nature of the composite Fermions by quantitatively
extracting this Berry phase. First we point out that longitudinal thermopower
(Seebeck effect) is non-vanishing due to the unusual nature of particle hole
symmetry in this context and is not sensitive to the Berry phase. In contrast,
we find that off-diagonal thermopower (Nernst effect) is directly related to
the topological structure of the composite Fermi surface, vanishing for zero
Berry phase and taking its maximal value for $\pi$ Berry phase. In contrast, in
purely electrical transport signatures the Berry phase contributions appear as
small corrections to a large background signal, making the Nernst effect a
promising diagnostic of the Dirac nature of composite fermions.
