M-theory in the Omega-background and 5-dimensional non-commutative gauge theory

The $Ω$-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 $Ω$-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 $Ω$-background. The operators on the M5 branes, in the $Ω$-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.

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