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But we have a problem: 
is not well-defined on local expressions.
is not well-defined on local expressions.We can define it “by hand”, just to be zero on local expressions:
Local functionals form a Lie superalgebra under 
; we define
as the differential of its homology complex
; we define
as the differential of its homology complexBut we have to pay the price:
With this artificial definitions of 
, the theorems about the variation of 
generally speaking fail. Such theorems are needed to prove the closedness of 
.
, the theorems about the variation of
generally speaking fail. Such theorems are needed to prove the closedness of .However, some things we can still count on. First of all:
still does not change under small deformations of 
This is because we restrict ourselves to non-anomalous theories. The nontrivial
dependence of on the choice of is the
effect of anomaly; it must be absent in any consistent theory.
on the choice of is the
effect of anomaly; it must be absent in any consistent theory.Moreover, when 
describes the deformation of (the “gauge fermion”) we can still use:
describes the deformation of (the “gauge fermion”) we can still use:
This would seem to be potentially dangerous, as we are using the “regularized” in the
formula for Lie derivative in Eq. (19). But the key point
is, that one can always (at least locally) choose such Darboux coordinates that only
depend on fields and not on antifields. With such coordinates, the effect of regularization
in our “homological” definition of does not matter.
in the
formula for Lie derivative in Eq. (19). But the key point
is, that one can always (at least locally) choose such Darboux coordinates that only
depend on fields and not on antifields. With such coordinates, the effect of regularization
in our “homological” definition of does not matter.
Anomalies appear as effect of Jacobian of a field redefinition. When
is chosen to only depend on fields, the manipulations needed to prove the closedness of
do not actually involve field redefinitions. Therefore there is no anomaly in the closedness of
However, while there are no anomalies in the closedness of , there are
potentially anomalies in the closedness of its base analogue 
.
Indeed, in the proof of closedness of , we need to act with
on 
. In bosonic string theory, as in any theory coming from BRST formalism, it is true that 
only depends on antifields — see Eq. (11) where 
.
Therefore, the effect of regularization should not cause anomalies. However, in the
pure spinor string involves both fields and antifields. Therefore:
, there are
potentially anomalies in the closedness of its base analogue .
Indeed, in the proof of closedness of , we need to act with
on . In bosonic string theory, as in any theory coming from BRST formalism, it is true that
only depends on antifields — see Eq. (11) where .
Therefore, the effect of regularization should not cause anomalies. However, in the
pure spinor string involves both fields and antifields. Therefore:
For the pure spinor string we need to also prove the non-anomalousness of the
symmetry generated by the 
-ghost. This is needed for the construction of the base form .
-ghost. This is needed for the construction of the base form .The case of pure spinor string is work in progress with R. Lipinski.