1. Introduction
In this paper, we define various characteristic classes on a super vector bundle over a superspace, so called super characteristic classes. We also propose the super Riemann-Roch formulas and the super Gauss-Bonnet formulas as its application. In contrast, it is justified the definition of the super characteristic classes by establishing those formulas. In [1] , we defined the super Chern classes with values in the super number
,
and we succeeded in applying the super ADHM construction of the super Yang-Mills instantons. But essentially the super Chern classes ought to take with values in an integer
. Meaning like it, we introduce the new definition of the super Chern classes with values in integer. In general, the characteristic classes consider that given the vector bundles it corresponds to some cohomology class of the base manifolds. Hence, we need the cohomology reflecting the properties of superspaces. Therefore, we will define the cohomology with respect to coefficient of the some finitely generated group, which is called the helicity group.
This article is organized as follows. After a brief sketch on the definition and examples of superspaces and its cohomology in Section 2 ([1] -[6] ), main result in this paper is that we define the Chern class, Chern character, Todd class, Pontrjagin class, Eular class,
-genus and L-genus as in the case of super category in Section 3. In Section 4, as an application, we have the Riemann-Roch type formula of super structure sheaf on the complex supercurves of dimension
with genus g. Moreover, it generalizes the structure sheaf to any super line sheaves. In particular, in the case of dimension
, with
supersymmetric structure, we obtain the Atiyah-Singer index type formula for any super line bundles. In Section 5, we attempt to define the helicity group and cohomology with respect to coefficient of the helicity group. In Section 6, we give the Gauss-Bonnet type formula on the complex supercurves of dimension
with genus g and the complex super projectve space of dimension
.
2. Supermanifolds
We will summarize the definitions here in order to establish terminology and notation ([1] -[6] ).
Definition 2.1 A superspace is defined to be a local ringed space
consisting a topological
space M and a sheaf of
-graded supercommutative rings
on it such that the stalk
at any point
is a local ring.
In particular case of a superspace, a supermanifold is defined by the following.
Definition 2.2 A supermanifold of dimension
is a ringed space
with the following properties:
1) the structure sheaf
is a sheaf of
-graded supercommutative rings,
2) Let
be the ideal sheaf of nilpotents in
. Then
is a classical manifold M of dimension n, so also called body.
3) Let
be the locally free
-module of rank
. Then
is locally isomorphic to the exterior algebra
.
A supermanifold is said to be split if the isomorphism 3) holds globally.
A local section
can be expressed as follows:
(1)
where
,
is a local coordinate function on
and
a local
generator of
. We refer to
as a local coordinate of a supermanifold
.
Example 2.1 1) The typical example is the real (or complex) linear superspace
(or
) which can be defined by
![]()
![]()
where
(or
) is the sheaf of the ring of differential functions on
(or
). It is easy to see that
the
is isomorphic to
.
2) A real super sphere of dimension
is defined by
![]()
where
is the sheaf of the ring of differential functions on
.
3) A complex super projective space of dimensin
is defined by
![]()
We denote by
the structure sheaf
of
. A super holomorphic function 1) on
should be a function of total homogeneity 0 in
even variables
and N odd variables
, that is
has homogeneity
. Let
be the even line sheaf of degree d on
and
be the odd line sheaf of degree d on
.
4) A quaternionic super projective space of dimension
is defined by
![]()
The above are examples of the supermanifolds in Definition 2.2.
5) We have a new example of superspace in Definition 2.1 as follows. The complex supercurves of dimension
with genus g is defined by
![]()
where
is the canonical line bundle on the classical Riemann surfaces
and
. In the case of
, it becomes the super Riemann surfaces with
SUSY structure (c.f. [7] , p.162). In the case of
, we do not kown whether or not there exists a SUSY structure.
We can construct the super Euler sequence as follows ([1] ).
![]()
Tensoring this with
, we have
![]()
Considering the super determinant ( so called Berezin bundle ) of the super Euler sequence, we obtain
![]()
Dualizing this, we can write
![]()
where
calls the canonical super line bundle of
and
is the parity change functor. The fol- lowing is given by Manin ([5] ).
Lemma 2.1
![]()
![]()
![]()
where
and
.
The following is given by Penkov ([8] ).
Theorem 2.1 (Super Serre Duality) Let E be a complex super vector bundle over
. Suppose that
is the canonical super line bundle of
. Then we have the following.
![]()
3. Super Characteristic Class
In this section, we will give a main result in this paper. Let
denote the structure sheaf on
. Then we have an exact sequence (cf. [2] , p.166 Lemma 2.1)
![]()
where
is the natural injection and exp is defined by
![]()
The
implies
,
. Hence
. This induces the exact sequence of cohomology groups:
![]()
We can identify
with the equivalence classes of
or
-super line bundles over
. Then we can define the super first Chern class of
-super line bundle L and
-super line bundle
by
![]()
Remark 3.1 Note that we can define
. We consider the line sheaf
over the complex super projective space
. This line sheaf is decomposed into
![]()
The super first Chern calss and the classical first Chern class denote by
and
, respectively. Then we have
![]()
![]()
Hence, we see that for the superline bundle L
![]()
We will propose the axiomatic definition of super Chern classes (cf. [1] [2] [9] -[15] ). We consider the category of complex
-super vector bundles over an
-superspace
.
Axiom 1 For each complex super vector bundle E over
and for each positive integer i, the i-th super Chern class
is given, and
.
We set
and call
the total super Chern class of E.
Axiom 2 (Naturality)
Let E be a complex super vector bundle over a superspace
and
a morphism of superspaces. Then
![]()
where
is the pull-back bundle over
.
Axiom 3 (Whitney sum formula)
Let
be complex line bundles of rank
or
and
be their Whitney sum. Then
![]()
Axiom 4 (Normalization)
We put
and
. Then it can be axiomatically as follows:
![]()
![]()
![]()
In order to explicitly define the super characteristic classes we need the splitting principle ([2] Proposition 3.7) as follows.
Proposition 3.1 (Bartocci, Bruzzo, Hernandez-Ruiperez) Let E be a complex
-super vector bundle over an
-supermanifold
. Then there exists a supermanifold
and a proper fibration
such that
1) The homomorphism
is injective.
2) The pull-back bundle
splits into a direct sum of even complex line bundles
of rank
and odd complex line bundles
of rank
:
![]()
![]()
We will explicitly give the super characteristic classes.
Definition 3.1 1) The total super Chern class
is defined by
![]()
2) The total super Chern character
is defined by
![]()
3) The super Todd class
is defined by
![]()
4) The super Eular class
is defined by
![]()
5) Let
be a real vector bundle of rank
. The i-th super Pontrjagin class
and the total super Pontrjagin class are defined by
![]()
![]()
6) The super
-genus
is defined by
![]()
7) The super L-genus
is defined by
![]()
We can consider that it is justified these definitions by the following (cf. [13] [14] ).
Lemma 3.1 The first few terms of
and
are given by the following.
![]()
![]()
![]()
Proof. Let E be a complex rank-
super vector bundle over a complex
-dimensional supermanifold
. Then, total super Chern class is written by
![]()
Hence, we have
![]()
![]()
![]()
The total super Chern character is written by
![]()
Hence we have
,
,
,
.
It is well-known thtat
![]()
![]()
Hence the total super Todd class is written by
![]()
Therefore we have
,
,
,
.
Then, they satisfy that
![]()
![]()
![]()
W
Lemma 3.2 The first few terms of
,
and
are given by the following.
![]()
![]()
![]()
Proof.
,
and
similarly form in the classical case. Therefore
and
are of same argument (cf. [13] ).
Let E be a complex rank-
super vector bundle over a complex
-dimensional supermanifold
. The total super Chern class is written by
![]()
Hence, we have
,
,
,
.
The total super Pontrjagin class is written by
![]()
Hence, we have
,
.
Then, they satisfy that
![]()
W
4. Riemann-Roch Type Formula
Let
be the complex supercurves with genus g, where
,
,
in Example 2.1 (5). Then the canonical super line bundle on
is explicitly written by
![]()
Hence we have
.
Note that for any object E and F the parity change functor
satisfies
![]()
In general, if
is a supermanifold, then its tangent bundle can be written by
(cf. [16] ). Hence we have
![]()
Using this decomposition, Euler number of
get
![]()
Note that
.
Theorem 4.1 Let
be the complex
-dimensional supercurves with genus g. Then, we have a Noether type formula as follows.
![]()
Proof. Let
be the genus on the classical Riemann surfaces and
be
the number of linear independent Dirac zero modes or harmonic spinors which is not topologically invariant.
The structure sheaf of the complex supercurves have decomposition
.
In the case of genus
, we have (cf. ([17] ))
![]()
In the case of genus
, it always satisfies
for any p. In the case of genus
, it satisfies
![]()
In the case of genus
, we have the following.
![]()
![]()
Note that equal of second make use of the classical Serre duality. Hence we obtain
![]()
In the case of genus
and
, we can prove similarly. W
Corollary 4.1 Let
be the complex
-dimensional supercurves with genus g. Then we have a Riemann- Roch type formula as follows.
![]()
where
is the fundamental homology class.
Proof.
![]()
From Theorem 4.1, this completes the proof of Corollary 4.1. W
The following Corollary essentially has been obtained by [18] . It needs the
supersymmetric structure on the
super Riemann surfaces (cf. [7] [19] [20] ). The following rewrite the result of [21] to the super characteristic classes.
Corollary 4.2 Let
be the complex
-dimensional super Riemann surfaces and
be any super line bundles of rank
on
. Then we have a Atiyah-Singer index type formula as follows.
![]()
where
is the fundamental homology class.
Proof. The canonical super line bundle
of a super Riemann surface
can be defined by splitting the Berezin bundle
using the super complex structure
. We get an exact sequence ([21] [22] )
![]()
We can define the operator
,
,
,
. Note that the
operator
is
supersymmetric anti-holomorphic vector fields. Tensoring this exact sequence with any super line bundles
, we have
![]()
We can define the operator
,
,
. We can describe ![]()
as the space of sections s of
satisfying the condition
. The group
can be described as
the space of sections
modulo the image of the operator
. Hence ![]()
and
. W
Let
be
distinct points and
,
. Then the super meromorphic functions
![]()
is coresponding to the super Weil divisor (cf. [18] [20] )
![]()
where
is a super holomorphic function. We put
,
,
and
. Then the inverse element of
, which is unique, is given by the formula
(cf. [23] ). As an application, we have a main theorem as follows.
Theorem 4.2 Let
be the complex
-dimensional supercurves with genus g and
be
any super line bundles of rank
on
. Then we have a Riemann-Roch type formula as follows.
![]()
where
is the fundamental homology class.
Proof. Let us consider the super divisor
on
. The local equation on D is defined by
on a open set
of
. If
, then
. The super Weil divisor can be considered as the super Cartier divisor. Then there is the exact sequence
![]()
The line sheaf
corresponding to
is defined by the transition functions
![]()
on
. The sheaf
which is defined by
![]()
is the coherent ideal sheaf. The fiber
of
is of zero in
and
in
. The sheaf
is called the super skyscraper sheaf. Tensoring this with
, we have
![]()
The map
is defined by
on an open set
of
. Taking co-
homology, this gives a long exact sequence
![]()
Taking the alternative sum, we have
![]()
Noting that
and
, we have
![]()
From
and
, we have
![]()
We also take the exact sequence
![]()
This gives rise to a long exact sequence
![]()
Taking also the alternative sum, we have
![]()
Hence, we havet
![]()
Note that
. So adding
in both side, we see that
![]()
Therefore,
is independent of
, so that we can put
.
From Theorem 6.1,
. This completes the proof of
Theorem 4.2. W
5. Helicity Group
Definition 5.1 The helicity rank of finitely generated group G is defined by the positive generator of linearly independent itself. The helicity rank is denoted by
. The helicity rank of
is defined by the negative generator of linearly independent itself. The helicity rank of
also is defined by twice the positive generator of linearly independent itself of G.
We define the finitely generated group of two type as follows.
![]()
![]()
Note that
,
and
are isomorphic to
,
and
as abelian groups, respectively.
But its helicity rank is differently as follows.
Example 5.1
,
,
,
, , , ,
, ,.
Definition 5.2 Let
be a
-dimensional complex supermanifold. Then the helicity group
is defined by the following.
![]()
The helicity rank of
can be represented by
![]()
The super cohomology with coefficient in
of an
-dimensional supermanifold
is isomorphic to the
-valued cohomology with coefficient in
of the classical manifold M using the universal coefficient theorem. That is to say, we have the following.
![]()
This isomrphism is applied in section 6.
6. Gauss-Bonnet Type Formula
In this section, we will apply the super cohomology with coefficient in helicity group
.
Theorem 6.1 Let
be the complex
-dimensional supercurves with genus g. Then we have a Gauss- Bonnet type formula as follows.
![]()
Proof. Euler number of
get
![]()
Note that
. On the other hand, the right hand side is
.
Both sides coincide. W
Theorem 6.2 Let
be the complex
-dimensional super projective space. Then, we have
![]()
Proof.
From the super Euler sequence, we can compute the total Chern class of holomorphic tangent bundle
. Setting
for simplicity’s sake and
, we have
![]()
The sum of coefficient of x is the first super Chern number
.
W.