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The main purpose of this article is to define the super characteristic classes on a super vector bundle over a superspace. As an application, we propose the examples of Riemann-Roch type formula. We also introduce the helicity group and cohomology with respect to coefficient of the helicity group. As an application, we propose the examples of Gauss-Bonnet type formula.

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 [

This article is organized as follows. After a brief sketch on the definition and examples of superspaces and its cohomology in Section 2 ([

We will summarize the definitions here in order to establish terminology and notation ([

Definition 2.1 A superspace is defined to be a local ringed space

space M and a sheaf of

In particular case of a superspace, a supermanifold is defined by the following.

Definition 2.2 A supermanifold of dimension

1) the structure sheaf

2) Let

3) Let

A supermanifold is said to be split if the isomorphism 3) holds globally.

A local section

where

generator of

Example 2.1 1) The typical example is the real (or complex) linear superspace

where

the

2) A real super sphere of dimension

where

3) A complex super projective space of dimensin

We denote by

4) A quaternionic super projective space of dimension

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

where

We can construct the super Euler sequence as follows ([

Tensoring this with

Considering the super determinant ( so called Berezin bundle ) of the super Euler sequence, we obtain

Dualizing this, we can write

where

Lemma 2.1

where

The following is given by Penkov ([

Theorem 2.1 (Super Serre Duality) Let E be a complex super vector bundle over

In this section, we will give a main result in this paper. Let

where

The

We can identify

Remark 3.1 Note that we can define

The super first Chern calss and the classical first Chern class denote by

Hence, we see that for the superline bundle L

We will propose the axiomatic definition of super Chern classes (cf. [

Axiom 1 For each complex super vector bundle E over

We set

Axiom 2 (Naturality)

Let E be a complex super vector bundle over a superspace

where

Axiom 3 (Whitney sum formula)

Let

Axiom 4 (Normalization)

We put

In order to explicitly define the super characteristic classes we need the splitting principle ([

Proposition 3.1 (Bartocci, Bruzzo, Hernandez-Ruiperez) Let E be a complex

1) The homomorphism

2) The pull-back bundle

We will explicitly give the super characteristic classes.

Definition 3.1 1) The total super Chern class

2) The total super Chern character

3) The super Todd class

4) The super Eular class

5) Let

6) The super

7) The super L-genus

We can consider that it is justified these definitions by the following (cf. [

Lemma 3.1 The first few terms of

Proof. Let E be a complex rank-

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

Proof.

Let E be a complex rank-

Hence, we have

The total super Pontrjagin class is written by

Hence, we have

Then, they satisfy that

W

Let

in Example 2.1 (5). Then the canonical super line bundle on

Hence we have

Note that for any object E and F the parity change functor

In general, if

Using this decomposition, Euler number of

Note that

Theorem 4.1 Let

Proof. Let

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

In the case of genus

In the case of genus

Note that equal of second make use of the classical Serre duality. Hence we obtain

In the case of genus

Corollary 4.1 Let

where

Proof.

From Theorem 4.1, this completes the proof of Corollary 4.1. W

The following Corollary essentially has been obtained by [

Corollary 4.2 Let

where

Proof. The canonical super line bundle

We can define the operator

operator

We can define the operator

as the space of sections s of

the space of sections

and

Let

is coresponding to the super Weil divisor (cf. [

where

Theorem 4.2 Let

any super line bundles of rank

where

Proof. Let us consider the super divisor

The line sheaf

on

is the coherent ideal sheaf. The fiber

The map

homology, this gives a long exact sequence

Taking the alternative sum, we have

Noting that

From

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

Therefore,

From Theorem 6.1,

Theorem 4.2. W

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

We define the finitely generated group of two type as follows.

Note that

But its helicity rank is differently as follows.

Example 5.1

Definition 5.2 Let

The helicity rank of

The super cohomology with coefficient in

This isomrphism is applied in section 6.

In this section, we will apply the super cohomology with coefficient in helicity group

Theorem 6.1 Let

Proof. Euler number of

Note that

Both sides coincide. W

Theorem 6.2 Let

Proof.

From the super Euler sequence, we can compute the total Chern class of holomorphic tangent bundle

The sum of coefficient of x is the first super Chern number