Craps-Skenderis trick
1 Dynamical punctures
The idea of
CrapsSkenderis
was to promote the points of insertion into dynamical degrees of freedom
.
The corresponding antifields will be denoted
.
This procedure extends the BV phase space by a finite-dimensional (or “discrete”) piece.
Let us define the half-density
on this extended BV phase space as follows:
2 Partially quantum Master Equation
Proof The terms in
relevant for computing
in this context are:
This implies:
3 Geometrical interpretation
3.1 Cartan formula for de Rham differential
The nilpotent operator:
acting on
is the cohomological operator of the algebroid
; its cohomology coincides with
the de Rham cohomology of
. This is true for any manifold
, in our case
is the two-dimensional string worldsheet.
The first term on the RHS of Eq. (
24) is the term
in
(see Eq. (
23)).
For the other terms in
we do not see a clear geometrical meaning.
3.2 Volume element
To have a minimal geometrical example similar to
we imagine that rather than a
function on
, our
be a volume element. We then define
by the formula
(we should discard the continuous part of
, but keep the “discrete” part).
In other words, instead of acting on functions we consider vector fields acting on volume densities.
TODO: Is this a generalization of the de Rham cohomology?
4 String amplitudes
We consider a family of Lagrangian submanifolds parameterized by
taking
The form
looks as follows:
After integration over
(and omitting indices
for brevity) we get:
where
stands for the traceless part of
.
This result is the standard expression for the string amplitude.
is diffeomorphism-invariant in the following sense:
is Weyl-invariant
-part of the odd Laplace operator
