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
2 What is symmetry?
An even vector field on 
is a symmetry if:
is a symmetry if:
It preserves
, i.e. is of the form 
, 
Lie derivative of
along it is zero:
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Since we assume that satisfies the Master Equation, this condition reduces to:
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.
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:
is nilpotent:
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This allows us to construct the symmetries very easily:
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We will restrict ourselves to only considering symmetries of this form.