BRST Quantization - Mathematical Approach To BRST

Mathematical Approach To BRST

BRST construction, applies to a situation of a hamiltonian action of a compact, connected Lie group on a phase space . Let be the Lie algebra of and a regular value of the moment map . Let . Assume the -action on is free and proper, and consider the space of -orbits on, which is also known as a Symplectic Reduction quotient .

First, using the regular sequence of functions defining inside, construct a Koszul complex . The differential, on this complex is an odd -linear derivation of the graded -algebra . This odd derivation is defined by extending the Lie algebra homomorphim of the hamiltonian action. The resulting Koszul complex is the Koszul complex of the -module, where is the symmetric algebra of, and the module structure comes from a ring homomorphism induced by the hamiltonian action .

This Koszul complex is a resolution of the -module, i.e.,

, if and zero otherwise.

Then, consider the Chevalley-Eilenberg cochain complex for the Koszul complex considered as a dg module over the Lie algebra :

The "horizontal" differential is defined on the coefficients by the action of and on as the exterior derivative of right-invariant differential forms on the Lie group, whose Lie algebra is .

Let be a complex such that with a differential . The cohomology groups of are computed using a spectral sequence associated to the double complex .

The first term of the spectral sequence computes the cohomology of the "vertical" differential :

, if and zero otherwise.

The first term of the spectral sequence may be interpreted as the complex of vertical differential forms for the fiber bundle .

The second term of the spectral sequence computes the cohomology of the "horizontal" differential on :

, if and zero otherwise.

The spectral sequence collapses at the second term, so that, which is concentrated in degree zero.

Therefore, if p = 0 and 0 otherwise.