Odd Laplace operator
Here we will give a brief self-contained presentation of the “canonical” odd Laplace operator .
Odd Laplace operator is closely related to the notion of Lie derivative; we will start by
discussing some properties of Lie derivative.
1 Properties 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 Definition 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:
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
:
Eq. (
4) is the definition of
(as was discovered in
Khudaverdian).
We will now prove Theorem 1.
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.
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
.
Proof
For any
:
Proof of Theorem 1
We can in any case
define by the formula:
What we have to prove is that:
| so defined does not depend on |
| | |
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:
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
.
3 Lie 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:
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 .)
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:
4 The canonical operator is nilpotent
Indeed, since the definition of
is geometrically natural, it automatically commutes with canonical
transformations and therefore for any
:
Comparing this with Eq. (
10) we derive:
5 Relation to odd Poisson bracket
We will define the operator
on functions as follows:
Usually there is some obvious implicit half-density; then we will abbreviate:
6 In coordinates
6.1 Leading symbol
The
leading symbol of
does not depend on
:
In Darboux coordinates (defined by Eq. (
1)):
6.2 The “quantum part”
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:
Obviously, this notation only makes sense with a choice of coordinates.
7 Example: purely even
As an example, consider the case when
has dimension
. A sufficiently generic
Lagrangian submanifold is given by the equation:
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
:
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).)