Noncommutative rigidity of the moduli stack of stable pointed curves

(joint work with Taro Sano)
It was proven by Paul Hacking that the Deligne-Mumford moduli stack of
stable n-pointed genus g curves is rigid (in characteristic zero).
In this talk we prove that the 2nd Hochschild cohomology of the moduli stack
is trivial except when (g, n) = (0, 5).
This implies a stronger result; namely, the rigidity of
the abelian category of coherent sheaves on the moduli stack.