Odd Laplace operator is closely related to the notion of Lie derivative; we will start by
discussing some properties of Lie derivative.
1Properties of Lie derivative of half-density
Consider a vector field on , and the corresponding 1-parameter group of diffeomorphisms .
Let us think of a half-density as a function of and , where is a point of
and a basis in , depending on in the following way:
By definition, the Lie derivative of along is:
Let us multiply by a function such that . The flux of preserves the point ,
and we have:
This implies that for any and :
In particular:
(2)
2Definition of
Let denote the BV phase space.
A -structure (i.e. an odd Poisson bracket on ) defines a canonical
second order differential operator on half-densities, which we will call
.
It is defined in the following way. Any half-density
defines a measure on a Lagrangian submanifold GeomBV,SemiClassical, which we will
denote , or sometimes simply just .
Given a smooth function , let us consider the variation of
under the variation of specified by the Hamiltonian vector field
corresponding to . It can only depend on the restriction of on (this
restriction may be called the “infinitesimal generating function” of the
variation of , or the “infinitesimal gauge fermion”). Therefore, this variation should be of the form:
(3)
where is some integral form on (which of course depends on ).
We will now argue that exists some half-density on ,
which we will denote , such that:
In other words:
Theorem 1:
given a half-density , exists another half-density
, such that for any and any Lagrangian :
(4)
Eq. (4) is the definition of
(as was discovered in Khudaverdian).
Lemma 2 Our
(which is a density on defined, given , by Eq. (3))
only depends on through restriction to the first infinitesimal neighborhood of . In
other words, if we replace with where is a function on
having second order zero on , then will not change.
This is slightly counterintuitive, because is actually a second order differential operator.
It is important that is Lagrangian.
Proof The definition of is given by Eq. (3); only enters the light hand side of
Eq. (3) through the first infinitesimal neighborhood of .
We will now prove that a function can locally be extended from a Lagrangian
submanifold into the BV phase space so that the Hamiltonian vector field of the
extended preserves . (This is only true locally.)
Lemma 3
For any point , a fixed positive integer , and a smooth
function on , exists an open neighborhood of , such that can be extended
from to a function on such that the derivative of along the flux of
has zero of the order on .
Proof Direct computation in coordinates. Let us choose some Darboux coordinates
, so that is at . Let us use these coordinates to identify half-densities with
functions. Without loss of generality, we can assume that in the vicinity of :
where stand for terms of the higher order in . Then our problem is to find
solving:
Solutions can always be found, order by order in , to any order .
For example, when :
where should satisfy:
This equation always has a solution in a sufficiently small neighborhood of .
Lemma 4
(5)
Proof
For any :
Proof of Theorem 1
We can in any case define by the formula:
(6)
What we have to prove is that:
so defined does not depend on
(7)
Consider any and a Lagrangian submanifold such that and
in such that are tangent to and .
Then, Eq. (6) says:
by definition
Let us consider Eq. (5) in the special case when is such that .
We get:
(8)
Consider an infinitesimal variation of specified by some “gauge fermion” .
Let us use Lemma 3 to extend it to , and put .
Lemma 2 implies that of the RHS of Eq. (8)
vanishes. This proves that the variation with respect to of the LHS of Eq. (8) vanishes, and
therefore does not depend on .
3Lie derivative in terms of
The purpose of this Section is to prove Eq. (10) “geometrically”. (In fact Eq. (10)
can be proven by a direct local computation in coordinates.)
Let us fix two functions and .
Let us suppose that is odd. Then:
For any Lagrangian submanifold , let us consider:
(9)
Consider the case when the restriction of to is zero. Then Eq. (9) implies that the
restriction of on such is equal to . We will use as a “test
function” and assume that has compact support, contained in a sufficiently small
open superdomain .
The submanifold given by the equation contains sufficiently many Lagrangian
submanifolds, in the following sense: if the restriction of a density on any Lagrangian
submanifold contained in is zero, then the density is zero everywhere on .
(If we were working with ordinary (not super) manifolds, we would say that through
every pointof passes at least one Lagrangian submanifold fully contained in .)
Indeed, when is small enough, we can consider the space of trajectories of on .
It is an odd symplectic manifold (the odd analogue of the Hamiltonian reduction). It has
sufficiently many Lagrangian submanifolds, in the above sense. They lift to
Lagrangian submanifolds in .
Therefore Eq. (9) implies that on : .
To extend this formula
from to the whole , let us consider the superdomain ; the fermionic
coordinate of will be denoted . Consider the subspace of given by the equation
. It has sufficiently many maximally isotropic submanifolds. Then the same
computation as in Eq. (9) gives:
where is some funcion on . But by definition does not depend on . Therefore
. This implies, for odd :
If instead of odd we consider some even , then this argument does not work,
because when there are no Lagrangian submanifolds
contained in level sets of . But, given some odd and a constant Grassmann parameter , we can apply
the argument to the odd Hamiltonian . Considering the coefficient of proves that for
even :
The formula which works for both even and odd is:
(10)
4The canonical operator is nilpotent
Indeed, since the definition of is geometrically natural, it automatically commutes with canonical
transformations and therefore for any :
Let us verify that transforms correctly when we go to different Darboux coordinates.
An infinitesimal change of Darboux coordinates is generated by a Hamiltonian flux ; for an infinitesimal :
Let us assume that is odd. We have:
(13)
On the other hand, the additional term which comes from varying as a density is:
Consider the case when is an odd cotangent bundle: . Let us introduce
the coordinates on . Let
be the corresponding dual coordinates in the fiber of .
The odd Poisson bracket is:
With respect to these coordinates, we can define the “constant” volume element:
and the constant half-density:
We will introduce the following notation:
(14)
Obviously, this notation only makes sense with a choice of coordinates.
7Example: purely even
As an example, consider the case when has dimension . A sufficiently generic
Lagrangian submanifold is given by the equation:
(15)
The integral of a half-density over this
Lagrangian submanifold is:
To compute the variation of this integral under the Hamiltonian vector field , we just have to take into account the variation of the Lagrangian submanifold (15),
which is encoded in the following variation of :
(16)
After integration by parts, we get:
Therefore, we have:
This is in agreement with Eqs. (4) and (12). (Remember that in this case , see Eq. (16).)