charge algebra

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Contour argument

Consider the contour integral of two holomorphic fields

$\oint_w\!dz\, a(z) b(w)\,$

around the point $w\,$ as shown in Fig 1. When we perform a path integral with these fields inserted, they become time ordered, or radially ordered. Note that we can split the contour into two contours that either enclose or avoid the point $w\,$ :

Fig 1. Contour around $w\,$

At this point we employ the holomorphicity of $a(z)\,$ and deform the contours into $C_1: |z| > |w|\,$ and $C_2: |z| < |w|\,$ as shown in Fig 2.

Fig 2. Contours with unambiguous time ordering $w\,$

We can then write

$\mathcal{R}\oint_w\!dz\, a(z) b(w) = \oint_{C_1}\!dz\, a(z) b(w) + \oint_{C_2}\!dz\, b(w) a(z)\,$ ,

where it is understood that this is an operator equation. We can further deform the contours into circles around the origin with radii $|w| \pm \epsilon\,$ , and write

$\mathcal{R}\oint_w\!dz\, a(z) b(w)\,$	$= \oint_{\|z\|=\|w\|+\epsilon}\!dz\, a(z) b(w) - \oint_{\|z\|=\|w\|-\epsilon}\!dz\, b(w)a(z)\,$ ,
	$= \left(\oint_{\|z\|=\|w\|+\epsilon}\!dz\, a(z)\right) b(w) - b(w) \left(\oint_{\|z\|=\|w\|-\epsilon}\!dz\, a(z)\right)\,$ ,
	$= [A, b(w)]\,$ ,

where we have switched the sense of $C_2\,$ to be counterclockwise, and where we have defined $A = \oint_0\!dz\, a(z)\,$ . Since $a(z)\,$ is holomorphic except at the origin, the integral can be performed around any circle. We can take $b(w)\,$ outside the integral due to additive associativity of the OPE. The interpretation is as follows: $a(z)\,$ is the holomorphic component of a conserved current, since $\bar\partial a(z) = 0\,$ , and to calculate $A\,$ we may perform an integral over space of $a(z)\,$ at constant time. Therefore $A\,$ is the charge associated with that current. If the current arises from some symmetry (due to Noether's theorem, then $A\,$ generates a symmetry transformation, and

$\delta_A b(w) = [A,b(w)]\,$ .

Further, writing $B = \int_0\!dw\, b(w)\,$ , we find that

$[A, B] = \oint_0\!dw\,\oint_w\!dz\, a(z) b(w)\,$ ,

where radial ordering is understood.

We should stress that the contour deformation relied on there being no singular terms between $C_1\,$ and $C_2\,$ . Thus we may extend the argument to a number of different fields besides $a(z)\,$ and $b(w)\,$ , as long as none of the other fields have a singular OPE with $a(z)\,$ lying between $C_1\,$ and $C_2\,$ . If this condition is satisfied, then by the residue theorem, only the term $1/(z-w)\,$ in the OPE between $a(z)\,$ and $b(w)\,$ contributes, and we obtain: