  The $\Omega$-background is defined for supergravity, and a very general class
of such backgrounds is constructed for 11-dimensional supergravity.
11-dimensional supergravity in a certain $\Omega$-background is shown to be
equivalent to a 5-dimensional non-commutative gauge theory of Chern-Simons
type. M2 and M5 branes are expressed as 1 and 2-dimensional extended objects in
the 5-dimensional gauge theory. This 5-dimensional gauge theory is shown to
admit a consistent quantization with two coupling constants, despite being
formally non-renormalizable.
  A check of a twisted version of AdS/CFT is performed relating this
5-dimensional non-commutative gauge theory to the theory on N M5 branes,
wrapping an $A_{k-1}$ singularity and placed in an $\Omega$-background. The
operators on the M5 branes, in the $\Omega$-background, are described by a
certain chiral algebra which in the large N limit becomes a $W_{k+\infty}$
algebra. This chiral algebra is recovered from the 5-dimensional gauge theory.
  This argument also provides a holographic explanation of the result of
Maulik-Okounkov and Schiffmann-Vasserot that the $W_{k+\infty}$ algebra acts on
the equivariant cohomology of the moduli of instantons on an $A_{k-1}$
singularity.
