Faddeev-Popov integration

Now we will consider a special case when tangent vectors to orbits of generate the entire , or, with our notations, . We will show that in this case the integration over the “rotated” Lagrangian submanifold of Eq. (3) is equivalent to integration over some family of Lagrangian submanifolds, more precisely over a family of sections of . We reproduce the Faddeev-Popov integration formula, Eq. (6)

Suppose that

is specified by the equation:

where

is some vector space. Consider the following family of Lagrangian submanifolds of

parametrized by

. For every

, the corresponding Lagrangian submanifold is given by the following section of

naturally associated to any

where

is a derivative map

We use to denote elements of in order to agree with the standard notations in BRST formalism. At this point it has nothing to do with the complex conjugate of the Faddeev-Popov ghost .

We therefore have a family of Lagrangian submanifolds parametrized by elements of the vector space

. The generating function of the infinitesimal variation of the Lagrangian submanifold from

Indeed:

Therefore the form

is given by:

The corresponding integral form is obtained by the Baranov-Schwarz transform:

Let us integrate

over the whole

(5)

This result coincides with the integration over the conormal bundle of

Given the explicit form of

, its restriction to

is:

(6)

This integral is a priori divergent, because we integrate over the noncompact space . Sometimes, it can be made convergent by a special choice of the integration contour — see the example of the Yang-Mills theory.

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