M-theory in the Omega-background and 5-dimensional non-commutative gauge theory
M-theory in the Omega-background and 5-dimensional non-commutative gauge theory
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Omega背景中的M理论和5维非交换规范理论
DOI：
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发表时间：
2016
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通讯作者：
K. Costello
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作者：
K. Costello
$\Omega$-背景是为超重力定义的，并且为 11 维超重力构造了此类背景的非常通用的类别。特定$\Omega$背景下的11维超引力被证明等价于Chern-Simons型的5维非交换规范理论。 M2和M5膜在5维规范理论中被表示为1维和2维扩展对象。尽管形式上是不可重整化的，但这种 5 维规范理论被证明允许具有两个耦合常数的一致量化。
对 AdS/CFT 的扭曲版本进行检查，将这个 5 维非交换规范理论与 N M5 膜上的理论联系起来，包裹 $A_{k-1}$ 奇点并放置在 $\Omega$-背景中。在 $\Omega$ 背景中，M5 膜上的算子由某个手征代数描述，该代数在大 N 极限下变成 $W_{k+\infty}$ 代数。这个手征代数是从五维规范理论中恢复出来的。
该论证还为 Maulik-Okounkov 和 Schiffmann-Vasserot 的结果提供了全息解释，即 $W_{k+\infty}$ 代数作用于 $A_{k-1}$ 奇点上瞬子模的等变上同调。
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.