On this page:
1 Why symmetries are necessary?
2 What is symmetry?
3 Construction of symmetries

Role of symmetries

1 Why symmetries are necessary?

The problem is:

does not have nontrivial integration cycles

In order to get nontrivial integration cycles:

we have to consider the factorspace of over some symmetry of

2 What is symmetry?

An even vector field on is a symmetry if:
  1. It preserves , i.e. is of the form ,

  2. Lie derivative of along it is zero:

Since we assume that satisfies the Master Equation, this condition reduces to:



This equation describes both symmetries and deformations of the action. When is odd, it corresponds to a symmetry; when is even, it corresponds to a deformation of the action.

3 Construction of symmetries

It is important that is nilpotent:

This allows us to construct the symmetries very easily:

  for any even is a symmetry of

We will restrict ourselves to only considering symmetries of this form.