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1 Restriction of symmetries
1.1 New operation
1.2 Definition of
1.3 Abelian
2 Proof of

Proof of equivariance

1 Restriction of symmetries

1.1 New operation

Symmetries are generated by Hamiltonians of the form: for arbitrary .

We want to restrict the possible to belong to some subspace

Let us introduce on functions on the operation as follows:

1.2 Definition of

Let us say that a subspace is admissible if:
  1. is closed under . This implies that is closed under

  2. For any exists such that:


  3. is super-antisymmetric. This actually implies that satisfies the Jacobi identity of a super Lie algebra; actually is a homomorphism of super Lie algebras:

  4. is an injection; in other words is isomorphic to

The requirement 3 almost follows from 1 and 2.

The requirement 4 is probably technical, but it is very convenient:
To any corresponds a symmetry generated by , and this correspondence is one-to-one

We will call the element of corresponding to , i.e. ,

1.3 Abelian

The simplest case is when is abelian, i.e. when for any and . In particular, this is the case for theories coming from BRST formalism. In this case it is very easy to solve Eq. (14):

   (for abelian )

However, not all abelian are admissible; we also have to check closedness under .

2 Proof of

The proof is based on the following algebraic interpretation of .

Consider the linear map:

[space of 1/2-densities on ]     

 [space of PDFs on ]

When satisfies the Master Equation, the RHS is our (usual, not equivariant). But we can evaluate this map on arbitrary , not necessarily satisfying the Master Equation

We observe that this map is an intertwiner between:

action of on half-densities in


action of on PDFs on

This is true even when . In particular, applying this intertwiner to the product in Eq. (7):