Heat equation for theta functions and vector-valued modular forms

We give a new method for constructing vector-valued modular forms from singular scalar-valued ones. As an application we prove the identity between two remarkable spaces of vector-valued modular forms which seem to be unrelated at a first look, since they are constructed in two very different ways. If $V_{grad}$ is the vector space generated by vector-valued modular forms constructed with gradients of odd theta functions and $V_{\mathrm{\Theta }}$ is the one generated by vector-valued modular forms arising from second order theta constants with our new construction, we will prove that $V_{grad}=V_{\mathrm{\Theta }}$. This result could also be proven as a consequence of the “heat equation” for theta functions.