Generalised Complex Contour Integral

We shall define the Generalised Complex Contour Integral as:

\[  \oint_X \Delta[\phi]D\phi \equiv i \int\int \delta[X]\left( \overline{\Delta}[\overline{\phi}]\det\left[\frac{\delta X}{\delta \phi}\right] + \Delta[\phi] \det\left[\frac{\delta X}{\delta \overline{\phi}}\right] \right) D\phi D\overline{\phi} \]

The function \[ \Delta[\phi] \]  is for \[ S_0 \] :


\[ \Delta[\phi]  = \frac{1}{2\pi i (\phi_1 - \phi_2)}\]


For \[S_n\] where n>0


\[ \Delta[\phi]  = \frac{1}{\int \varepsilon_{abc..}\phi^a(x) \partial_1 \phi^b(x) \partial_2 \phi^c(x)...     dx^n}\]


 


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