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Being motivated by applications to the physics of Weyl semimetals we studyspectral geometry of Dirac operator with an abelian gauge field and an axialvector field. We impose chiral bag boundary conditions with variable chiralphase $\theta$ on the fermions. We establish main properties of the spectralfunctions which ensure applicability of the $\zeta$ function regularization andof the usual heat kernel formulae for chiral and parity anomalies. We developcomputational methods, including a perturbation expansion for the heat kernel.We show that the terms in both anomalies which include electromagneticpotential are independent of $\theta$.

Heat Kernel, Spectral Functions and Anomalies in Weyl Semimetals