achieved, in the sense of a proof of the well definiteness of the functional integrals removing cut-offs; this would be parallel to [@DZ], in which the restriction to the bosonic sector of an hyperbolic sigma model coming by a disordered electron system was constructed.

The plan of the paper is the following. In §2 we define the model and we explain its supersymmetric representation. In §3 we analyze the critical theory at $E=0$, we derive Ward Identities and we show the validity of the Adler-Bardeen theorem and the supersymmetric cancellation of the anomalies in the limit of removed ultraviolet momentum cut-off; also, the relation with universality will be explained. In §4 we consider the non critical theory $E\not=0$ and we discuss the infrared problem. Finally, in §5 the main conclusions are discussed.

Disordered Chiral fermions and Supersymmetric representation
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The Dirac equation with vector disorder
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The (regularized) first quantized Hamiltonian describing chiral disordered Dirac fermions is H=\_[i=1]{}\^2 \_i (i\_[i]{}+g A\_i())\[11\] with $\xx=(x_1,x_2)\in \L_a$, $\L_a$ is a square lattice with step $a$ with antiperiodic boundary conditions, $\s_i$ are Pauli matrices $$\s_1=\begin{pmatrix} 0  & 1 \\ 1\0 &  0 \end{pmatrix}\quad\quad \s_2=
\begin{pmatrix} 0  & -i\\ i\0 &  0 \end{pmatrix}\quad\quad \s_0=
\begin{pmatrix} 